The slope of a line is 1.

Assertion (A): Its x-intercept and y-intercept are equal.

Reason (R): It makes an isosceles triangle with the coordinate axes Given below are the equation of two lines: y = 2x + 8 and y = 1/2x-7

Let's analyze the assertion (A) and the reason (R) in the context of the slope and intercepts of a line.

### Assertion (A): The x-intercept and y-intercept are equal if the slope of the line is 1.
- The general equation of a line in slope-intercept form is \( y = mx + c \), where \( m \) is the slope and \( c \) is the y-intercept.
- If the slope \( m = 1 \), the equation becomes \( y = x + c \).
- To find the x-intercept, set \( y = 0 \):
  \[
  0 = x + c \implies x = -c
  \]
- Therefore, the x-intercept is \( -c \) and the y-intercept is \( c \). For the x-intercept and y-intercept to be equal, \( -c = c \), which implies \( c = 0 \). This means the line passes through the origin (0, 0), and both intercepts are zero. In this specific case, the assertion is true only if the intercepts are zero.

### Reason (R): The line makes an isosceles triangle with the coordinate axes.
- If the line has a slope of 1 and does not pass through the origin, it will indeed form a right triangle with the x-axis and y-axis, where the legs (the intercepts) will be equal in length. This is the definition of an isosceles triangle.
  
However, this reasoning is general and does not necessarily apply to the specific case when both intercepts are zero.

### Analysis of the Given Equations
1. **Equation 1:** \( y = 2x + 8 \)
   - Slope \( m = 2 \), which is not 1. Therefore, it does not fit the assertion's criteria.
2. **Equation 2:** \( y = \frac{1}{2}x - 7 \)
   - Slope \( m = \frac{1}{2} \), which is also not 1. Therefore, it does not fit the assertion's criteria.

### Conclusion:
- The assertion (A) is **false** in the general case because the x-intercept and y-intercept are not necessarily equal unless both are zero.
- The reason (R) is **correct** for lines with a slope of 1 that form an isosceles triangle with the axes. However, it does not imply the correctness of (A) for all cases.

Thus, even though (R) is true, it does not correctly explain (A). The assertion is false in general, and the reason is true but irrelevant to the assertion.

Would you like further details or have any other questions?

Here are 5 questions related to this topic:
1. How do you find the x-intercept and y-intercept of a line from its equation?
2. What is the significance of the slope in the geometry of a line?
3. Can two different lines have the same slope? What does this imply?
4. How does changing the slope of a line affect its angle with the x-axis?
5. What are the conditions for two lines to be perpendicular?

**Tip:** When working with lines, always pay attention to both the slope and intercepts, as they provide crucial information about the line's behavior and position on the coordinate plane.

Explore the relationship between line slopes and intercepts in coordinate geometry. This page discusses the assertion that the x-intercept and y-intercept are equal when the slope of a line is 1, along with examples of lines with different slopes. Suitable for Grades 9-12.

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